arXiv subjects

Angela Pistoia

Publications and source records attributed to Angela Pistoia.

At least 19 recordsLinked to original sources

arXiv2026

Sign-changing multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions

We construct families of sign-changing solutions for the four-dimensional Brezis--Nirenberg problem \[ -\Delta u=u^3+\varepsilon u\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega, \] as $\varepsilon\to0^+$. A Lyapunov--Schmidt reduction shows that the location and relative scales of the bubbles are governed by a signed Green--Robin interaction matrix. We formulate an abstract existence criterion in terms of a simple positive eigenvalue admitting a positive eigenvector and a stable critical set. We then apply it to a positive--negative pair in a general domain and to several symmetric multi-peak configurations, including alternating regular polygons, orthogonal polygons, one central peak surrounded by peaks of the opposite sign, and aligned three-, four-, and five-peak patterns. For the two-peak solution we also prove that it has exactly two nodal domains and, under a natural balance condition and connectedness of the boundary, that the closure of its nodal set meets the boundary.

math.AP↗

arXiv2026

A fractional critical problem in the halfspace with Neumann conditions

Given $s\in(0,1)$ and $n>2s$, we construct nontrivial solutions of the problem \begin{equation*} \begin{cases} (-\Delta)^s u=u^{2^*_s-1} &{\mbox{ in }}\mathbb{R}^n_+,\\ {\mathcal{N}}_s u=0&{\mbox{ in }}\mathbb{R}^n_-,\end{cases} \end{equation*}where ${\mathcal{N}}_s$ represents the nonlocal Neumann condition of exterior type.

math.AP↗

arXiv2026

Sign-changing solutions to the Yamabe problem on a spherical cap

Spherical caps play a crucial role in establishing a criterion for the existence of solutions to the Yamabe problem on a compact Riemannian manifold with boundary, similar to the role played by the standard sphere in the problem on a closed Riemannian manifold. This problem is expressed in terms of a nonlinear boundary-value problem, where both the nonlinearity and the boundary condition are critical in the Sobolev sense. This work focuses on the existence of multiple solutions to the Yamabe problem on spherical caps. We show that if the spherical cap is contained in a hemisphere of the standard $n$-sphere and $n = 5$ or $n \geq 7$, the Yamabe problem has infinitely many sign-changing solutions. Our approach takes advantage of symmetries and is based on a careful analysis of the loss of compactness of the variational problem.

math.AP↗

arXiv2026

Sign-changing solutions to the Yamabe problem on manifolds with boundary

Let $(M, g)$ be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary is equivalent, in analytic terms, to finding a positive solution to a nonlinear boundary-value problem with critical growth. While the existence of positive solutions to this problem is by now well understood, the existence of sign-changing (nodal) solutions remains largely open. In this work we establish the existence of least-energy sign-changing solutions when the manifold is positive and the mean curvature of the boundary is a non-negative constant. More precisely, we prove that if $n\ge7$ and $M$ has a nonumbilic boundary point, then the problem admits least-energy nodal solutions. Our approach is variational and relies on the analysis of suitable conformal invariants and sharp energy estimates.

math.DG↗

arXiv2026

Blowing-up solutions to a critical 4D Neumann system in a competitive regime

We build blowing-up solutions to the critical elliptic system with Neumann boundary condition, \begin{equation*} \begin{cases} -Δu_1 + λu_1 = u_1^{3} -βu_1u_2^2 & \text{in } Ω, -Δu_2 + λu_2 = u_2^{3} -βu_1^2u_2 & \text{in } Ω, \frac{\partial u_1}{\partialν} = \frac{\partial u_2}{\partialν} = 0, & \text{on } \partial Ω, \end{cases} \end{equation*} when $λ>0$ is sufficiently large in a competitive regime (i.e. $ β>0$) and in a domain $Ω\subset\mathbb R^4$ with smooth protrusions.

math.AP↗

arXiv2026

Non-degeneracy of the bubble in a fractional and singular 1D Liouville equation

We prove the non-degeneracy of solutions to a fractional and singular Liouville equation defined on the whole real line in presence of a singular term. We use conformal transformations to rewrite the linearized equation as a Steklov eigenvalue problem posed in a bounded domain, which is defined either by an intersection or a union of two disks. We conclude by proving the simplicity of the corresponding eigenvalue.

math.AP↗

arXiv2026

On the simplicity of the sloshing eigenvalues

This paper investigates sloshing problems defined by $-Δu=0$ in $Ω$, with mixed boundary conditions: $\partial_νu=λu$ on $S$, and either $\partial_νu=0$ or $u=0$ on $W$. Here, $Ω$ represents a smooth bounded domain in $\mathbb{R}^n$ with boundary $\partialΩ=S \cup W$. We demonstrate that under small domain perturbations, all resulting eigenvalues are simple.

math.AP↗

arXiv2026

Entire solutions to a strongly competitive nonlinear Schrödinger system

We build infinitely-many non-radial positive solutions to the Schrödinger system \begin{equation*} \left\{\begin{aligned} &-Δu_1+u_1=u_1^{{\mathfrak p} }-Λu_1^{a_1} u_2^{a_2}\ \hbox{in}\ \mathbb R^N\\ &-Δu_2+u_2=u_2^{{\mathfrak p} }-Λu_1^{b_1}u_2^{b_2} \ \hbox{in}\ \mathbb R^N\\ \end{aligned}\right. \end{equation*} with sub-critical $\mathfrak p$-growth as $Λ\to +\infty$. The profile of each component is the sum of several copies of the positive solution to $-ΔU+U=U^{{\mathfrak p} }$ in $\mathbb R^N$, centered at suitable {\em peaks} whose mutual distances diverge as $Λ$ increases. More precisely, given two concentric regular polygons with $k$ sides and very large radii, the peaks of the first component are arranged along the edges of the {\em outer} polygon, alternated with those of the second component, and along the $k$ rays joining the vertices of the two polygons. To the best of our knowledge, this provides the first example of non-radial positive solutions for strongly competitive Schrödinger systems in the whole space.

math.AP↗

arXiv2026

Simplicity of eigenvalues for elliptic problems with mixed Steklov-Robin boundary condition

This paper investigates the spectral properties of two classes of elliptic problems characterized by mixed Steklov-Robin boundary conditions. Our main objective is to prove that, for a generic domain, all the eigenvalues are simple. This result is established by employing domain perturbation techniques and analyzing the transversality of the associated operators.

math.AP↗

arXiv2025

Blowing-up solutions to competitive critical systems in dimension 3

We study the critical system of $m\geq 2$ equations \begin{equation*} -Δu_i = u_i^5 + \sum_{j = 1,\,j\neq i}^m β_{ij} u_i^2 u_j^3\,, \quad u_i \gneqq 0 \quad \mbox{in } \mathbb{R}^3\,, \quad i \in \{1, \ldots, m\}\,, \end{equation*} where $β_{κ\ell} =α\in\mathbb{R}$ if $κ\neq\ell$, and $β_{\ell m}=β_{m κ} =β<0$, for $ κ, \ell \in \{1,\ldots, m-1\}$. We construct solutions to this system in the case where $β\to-\infty$ by means of a Ljapunov-Schmidt reduction argument. This allows us to identify the explicit form of the solution at main order: $u_1$ will look like a perturbation of the standard radial positive solution to the Yamabe equation, while $u_2$ will blow-up at the $k$ vertices of a regular planar polygon. The solutions to the other equations will replicate the blowing-up structure under an appropriate rotation that ensures $u_i\neq u_j$ for $i\neq j$. The result provides the first almost-explicit example of non-synchronized solutions to competitive critical systems in dimension 3.

math.AP↗

arXiv2025

Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity

We consider a nonautonomous semilinear elliptic problem where the power nonlinearity is multiplied by a discontinuous coefficient that equals one inside a bounded open set $Ω$ and it equals minus one in its complement. In the slightly subcritical regime, we prove the existence of concentrating positive and nodal solutions. Moreover, depending on the geometry of $Ω$, we establish multiplicity of positive solutions. Finally, in the critical case, we show the existence of a blow-up positive solution when $Ω$ has nontrivial topology. Our proofs rely on a Lyapunov-Schmidt reduction strategy which in these problems turns out to be remarkably simple. We take this opportunity to highlight certain aspects of the method that are often overlooked and present it in a more accessible and detailed manner for nonexperts.

math.AP↗

arXiv2025

A note on the first Steklov eigenvalue on planar domains

We consider the first positive Steklov eigenvalue on planar domains. First, we provide an example of a planar domain for which a first eigenfunction has a closed nodal line. Second, we establish a lower bound for the first positive eigenvalue on certain symmetric domains and show that this eigenvalue is simple for all ellipses. These results complement two statements contained in a work by Kuttler and Sigillito (Proc. Amer. Math. Soc. 20, 1969).

math.AP↗

arXiv2025

Segregated solutions for a class of systems with Lotka-Volterra interaction

This paper deals with the existence of positive solutions to the system $$ -Δw_1 - \varepsilon w_1 = μ_{1} w_1^{p} + βw_1 w_2\ \text{in } Ω,\ -Δw_2 - \varepsilon w_2 = μ_{2} w_2^{p} + βw_1 w_2 \ \text{in } Ω,\ w_1 = w_2 = 0 \ \text{on } \partial Ω, $$ where $Ω\subseteq \mathbb{R}^{N}$, $N \ge 4$, $ p ={N+2\over N-2}$ and $ \varepsilon $ is positive and sufficiently small. The interaction coefficient $ β= β(\varepsilon) \to 0 $ as $ \varepsilon \to 0 $. We construct a family of segregated solutions to this system, where each component blows-up at a different critical point of the Robin function as $\varepsilon \to 0. The system lacks a variational formulation due to its specific coupling form, which leads to essentially different behaviors in the subcritical, critical, and supercritical regimes and requires an appropriate functional settings to carry out the construction.

math.AP↗

arXiv2025

Sign-changing solutions for critical Hamiltonian systems in $\mathbb{R}^N$

We build infinitely many geometrically distinct non-radial sign-changing solutions for the Hamiltonian-type elliptic systems $$ -Δu =|v|^{p-1}v\ \hbox{in}\ \mathbb{R}^N,\ -Δv =|u|^{q-1}u\ \hbox{in}\ \mathbb{R}^N,$$ where the exponents $(p,q)$ satisfy $p,q>1$ and belong to the critical hyperbola $$\frac1{p+1}+\frac1{q+1} =\frac {N-2}N.$$ To establish this result, we introduce several new ideas and strategies that are both robust and potentially applicable to other critical problems lacking the Kelvin invariance.

math.AP↗

arXiv2025

Normalized vector solutions of nonlinear Schrödinger systems

Given $μ>0$ we look for solutions $ λ\in\mathbb{R}$ and $v_1,\dots,v_k\in H^1(\mathbb{R}^N)$ of the system \[ \begin{cases} \displaystyle -Δv_i+ λv_i+V_i(x)v_i = \sum_{\substack{j=1}}^kβ_{ij} v_iv_j^2 &\text{ in } \mathbb{R}^N, \text{ } i=1,\dots,k,\newline \displaystyle \int_{\mathbb{R}^N} \left(v_1^2+\dots+v_k^2 \right)\mathrm{d} x = μ, \end{cases}\] where $N=1,2,3$, $V_i:\mathbb R^N\to \mathbb R$ and $β_{ij}\in\mathbb{R}$ satisfy $β_{ij}=β_{ji}$ and $β_{ii}>0$. Under suitable assumptions on the $β_{ij}$'s, given a non-degenerate critical point $ξ_0$ of a suitable linear combination of the potentials $V_i$, we build solutions whose components concentrate at $ξ_0$ as the prescribed global mass $μ$ is either large (when $N=1$) or small (when $N=3$) or it approaches some critical threshold (when $N=2$).

math.AP↗

arXiv2025

New solutions for the Lane-Emden problem on planar domains

We consider the Lane-Emden problem on planar domains. When the exponent is large, the existence and multiplicity of solutions strongly depend on the geometric properties of the domain, which also deeply affect their qualitative behavior. Remarkably, a wide variety of solutions, both positive and sign-changing, have been found when the exponent is sufficiently large. In this paper, we focus on this topic and fine new sign-changing solutions that exhibit an unexpected concentration phenomenon as the exponent approaches infinity.

math.AP↗

arXiv2025

Multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions

The paper addresses the existence of multi-bubble solutions for the well-known Brezis-Nirenberg problem. Although there is extensive literature on the subject, the existence of solutions that blow up at multiple points in a 4D bounded domain remains an open problem. The goal of the present paper is to resolve this longstanding issue. In particular, we exhibit examples of domains where a large number of multi-bubble solutions exist. Our result can also be seen as the counterpart of the asymptotic analysis carried out by Konig and Laurin in Ann. Inst. H. Poincarè C Anal. Non Linèaire, 2024.

math.AP↗

arXiv2025

Concentrating solutions of nonlinear Schrödinger systems with mixed interactions

In this paper we study the existence of solutions to nonlinear Schrödinger systems with mixed couplings of attractive and repulsive forces, which arise from the models in Bose-Einstein condensates and nonlinear optics. In particular, we build solutions whose first component has one bump and the other components have several peaks forming a regular polygon around the single bump of the first component.

math.AP↗